Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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4
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95 views

Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question: Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$: $$ \frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t), $$ with smooth initial data ...
3
votes
0answers
42 views

Hopf Lemma for strong solutions

The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and ...
2
votes
0answers
41 views

About the rate for one approximation to the heat kernel

Let $G(t,x)$ be the heat kernel $$ G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}. $$ Here is one approximation of $G(t,x)$: $$ G_\epsilon(t,x)=e^{-t/\epsilon} \sum_{k=1}^\infty ...
2
votes
0answers
67 views

Stabilization of solution to one-dimensional system of PDE

I am trying to solve numerically next PDE system: $$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial ...
2
votes
0answers
97 views

Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
2
votes
0answers
96 views

Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
2
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0answers
123 views

regularity for viscosity solutions of second order parabolic equations

I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre . Here F is ...
1
vote
0answers
46 views

A comparison principle for degenerate parabolic equation

Let $\Omega$ be a bounded smooth domain and let $p < 0$ be real. Suppose that $u, v \in L^2(0,T;H^1(\Omega) \cap L^2(0,T;L^2(\partial\Omega))$ with $|u|^{p+1} \in L^2(0,T;L^2(\partial\Omega))$ and ...
1
vote
0answers
57 views

Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...
1
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0answers
45 views

Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...
1
vote
0answers
106 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
1
vote
0answers
87 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
1
vote
0answers
76 views

Proving solution to linear parabolic PDE is convex with negative third derivative

I have a PDE in $g(y,t)$ of the form \begin{equation} a\frac{\partial^2g}{\partial y^2}y^2-b\frac{\partial g}{\partial y}y -rg + \frac{\partial g}{\partial t} - c = 0 \end{equation} in which $a$, $b$, ...
1
vote
0answers
107 views

Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations singular at some point? For example, I am analyzing a partial differential equation $$ ...
1
vote
0answers
57 views

monotone parabolic systems, convex variational structure and Legendre transform

The context: for my research I am currently looking at parabolic systems of the type $$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
1
vote
0answers
174 views

local existence for a singular quasilinear parabolic equation

I'm considering the following type of PDE: $u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$ with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, ...
1
vote
0answers
56 views

Semiflows and continuous symmetries

Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\alpha F(u) = ...
0
votes
0answers
53 views

Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...
0
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0answers
58 views

Weak convergence of 4-th degrees

Good day! We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators. $u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...
0
votes
0answers
47 views

Interpolation with time continuity

If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE". Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in ...
0
votes
0answers
92 views

A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form $$u_t(t) - ...
0
votes
0answers
43 views

Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
0
votes
0answers
204 views

What is the solution of $u_t=u_{xx}+\frac{1}{x}u_x$?

What does the solution $u(x,t)$ of $u_t=u_{xx}+\frac{1}{x}u_x$ on $[0,1]$ with the following initial condition look like? $u(\frac{1}{2},0)=\delta(\frac{1}{2})$ (i.e. delta function at ...
0
votes
0answers
140 views

Smoothness of a solution for degenerate pde

I am looking at the IBVP: $$ u_t+x^2u_{xx}+u_y+f(x,y)=0, x,y \in D\in [0,\infty)\times[0,\infty),t\in(0,T]\\ $$$$ u(x,y,0)=F(x,y)\in C^{1,1}\\ $$$$ u(x,y,t)=0, x,y\in \partial D $$ and I would like to ...
0
votes
0answers
362 views

What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?